Propositional and First-Order Logic

Propositional logic focuses on propositions, which are statements that can only be classified as true or false. This logic systematically allows us to connect these propositions using logical operators, creating more complex expressions. The basic elements of propositional logic include cells like atomic propositions which represent the simple statements (for example, 'P' stands for 'It is raining'). Each proposition can independently hold a truth value - it is either true or false.

The syntax of propositional logic includes well-defined symbols and structures. Atomic propositions are the simplest expressions, represented by letters like P, Q, etc. Logical connectives are symbols that connect these propositions to form complex statements. The key connectives are: '¬' (not), '∧' (and), '∨' (or), '→' (implies), and '↔' (if and only if). Combining these connectives allows us to construct compound propositions, giving us the capacity to express a broader range of ideas.

Semantics in propositional logic refers to the meaning behind the propositions and the rules that determine their truth values. Each proposition is assigned a truth value, which can either be 'true' (T) or 'false' (F). The truth value of complex propositions can be evaluated based on the truth values of their constituent propositions and the logical connectives used.

Inference methods are techniques used to derive new propositions from existing ones. One common method is the truth table, which exhaustively lists all possible truth values for a set of propositions and determines the resulting truth values for complex expressions. Resolution is a rule of inference that allows for the elimination of contradictions among propositions. Forward chaining establishes conclusions from known facts while backward chaining starts with a goal and works backward to see if it can be reached with existing information.

While propositional logic is straightforward, it has limitations. It cannot capture complex relationships between objects or express quantified statements such as 'some' or 'all'. For example, propositional logic cannot express notions like 'All birds can fly' or 'Some cats are black' because it lacks the ability to quantify and relate different entities directly.

First-Order Logic (FOL) builds upon the foundations of propositional logic by adding elements that allow for a richer expression of knowledge. In FOL, we can use variables to represent objects, and quantifiers to express generalized or specific claims about these objects. Predicates define properties of objects or relationships between them. For instance, 'Loves(x, y)' can denote a relationship where one object loves another. This makes FOL more powerful and versatile for expressing complex scenarios and logical relationships.

This example illustrates FOL's capacity to make universal claims and deduce specific facts. The expression ∀x (Human(x) → Mortal(x)) states that if something is human, it is mortal. Next, by asserting 'Human(Socrates)', we identify Socrates as human. Through logical inference, we conclude that 'Mortal(Socrates)' must also be true, demonstrating how one can derive specific instances from general rules.

The power of First-Order Logic lies in its ability to capture complex relationships and express generalizations about objects and their properties. This enables FOL to handle intricate domains where simple true/false distinctions are inadequate. It's particularly useful in contexts such as mathematics, where relationships between numbers must be described, or in natural language processing, where meaning needs to be derived from sentences and phrases.